Question Statement

Evaluate the integral:

$\int cos θ (tan θ + sec θ) d θ$

Background and Explanation

This problem requires converting $tan θ$ and $sec θ$ to their sine-cosine equivalents, then simplifying the algebraic expression before applying basic integration rules for trigonometric functions.

Solution

Let $I$ represent the given integral:

$I = \int cos θ (tan θ + sec θ) d θ$

We begin by substituting the fundamental identities $tan θ = \frac{s i n θ}{c o s θ}$ and $sec θ = \frac{1}{c o s θ}$ to rewrite the integrand in terms of sine and cosine:

$I = \int cos θ (\frac{s i n θ}{c o s θ} + \frac{1}{c o s θ}) d θ$

Next, combine the fractions inside the parentheses over the common denominator $cos θ$ :

$I = \int cos θ (\frac{s i n θ + 1}{c o s θ}) d θ$

The $cos θ$ in the numerator (outside the parentheses) cancels with the $cos θ$ in the denominator:

$I = \int (sin θ + 1) d θ$

Using the linearity property of integrals, we split this into two separate integrals:

$I = \int sin θ d θ + \int 1 d θ$

Finally, we evaluate each integral using standard rules. The integral of $sin θ$ is $- cos θ$ , and the integral of $1$ with respect to $θ$ is $θ$ :

$I = - cos θ + θ + c$

where $c$ is the constant of integration.

Key Formulas or Methods Used

$tan θ = \frac{sin θ}{cos θ}$ (Tangent identity)
$sec θ = \frac{1}{cos θ}$ (Secant identity)
$\int sin θ d θ = - cos θ + C$ (Standard trigonometric integral)
$\int d θ = θ + C$ (Power rule for integration)
Linearity of integration: $\int [f (θ) + g (θ)] d θ = \int f (θ) d θ + \int g (θ) d θ$

Summary of Steps

Substitute identities: Replace $tan θ$ with $\frac{s i n θ}{c o s θ}$ and $sec θ$ with $\frac{1}{c o s θ}$
Combine fractions: Merge the terms in parentheses into a single fraction $\frac{s i n θ + 1}{c o s θ}$
Cancel terms: Multiply by $cos θ$ and cancel to obtain $sin θ + 1$
Split the integral: Separate into $\int sin θ d θ$ and $\int 1 d θ$
Integrate: Apply standard integration formulas to each term
Add constant: Combine results and include the arbitrary constant $c$

Solution

Let

I

represent the given integral:

I = \int cos θ (tan θ + sec θ) d θ

We begin by substituting the fundamental identities

tan θ = \frac{s i n θ}{c o s θ}

and

sec θ = \frac{1}{c o s θ}

to rewrite the integrand in terms of sine and cosine:

I = \int cos θ (\frac{s i n θ}{c o s θ} + \frac{1}{c o s θ}) d θ

Next, combine the fractions inside the parentheses over the common denominator

cos θ

I = \int cos θ (\frac{s i n θ + 1}{c o s θ}) d θ

The

cos θ

in the numerator (outside the parentheses) cancels with the

cos θ

in the denominator:

I = \int (sin θ + 1) d θ

Using the linearity property of integrals, we split this into two separate integrals:

I = \int sin θ d θ + \int 1 d θ

Finally, we evaluate each integral using standard rules. The integral of

sin θ

- cos θ

, and the integral of

1

with respect to

θ

θ

I = - cos θ + θ + c

where

c

is the constant of integration.

Summary of Steps

Substitute identities: Replace

tan θ

with

\frac{s i n θ}{c o s θ}

and

sec θ

with

\frac{1}{c o s θ}

Combine fractions: Merge the terms in parentheses into a single fraction

\frac{s i n θ + 1}{c o s θ}

Cancel terms: Multiply by

cos θ

and cancel to obtain

sin θ + 1

Split the integral: Separate into

\int sin θ d θ

and

\int 1 d θ

Integrate: Apply standard integration formulas to each term

Add constant: Combine results and include the arbitrary constant

c

3.2 Q-14

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

3.2 Q-14

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps